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Giải bài 17 trang 94 sách bài tập toán 8 - Cánh diều

Cho tam giác \(ABC\) có các đường trung tuyến \(BD\) và \(CE\). Lấy các điểm \(H,K\) sao cho \(E\) là trung điểm của \(CH,D\) là trung điểm của \(BK\). Chứng minh:

💃Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho tam giác \(ABC\) có các đường trung tuyến \(BD\) và \(CE\). Lấy các điểm \(H,K\) sao cho \(E\) là trung điểm của \(CH,D\) là trung điểm của \(BK\). Chứng minh: a)     Các tứ giác \(AHBC,AKCB\) là hình bình hành; b)    \(A\) là trung điểm của \(HK\).

Phương pháp giải - Xem chi tiết

Dựa vào dấu hiệu nhận biết của hình bình hành: -         Tứ giác có hai cặp cạnh đối bằng nhau là hình bình hành -         Tứ giác có hai đường chéo cắt nhau tại trung điểm của mỗi đường là hình bình hành -         Tứ giác có hai cặp góc đối bằng nhau là hình bình hành -         Tứ giác có hai cạnh đối song song và bằng nhau là hình bình hành.

Lời giải chi tiết

a)     Tứ giác \(AHBC\) có \(E\) là trung điểm của hai đường chéo \(AB\) và \(CH\) nên \(AHBC\) là hình bình hành.Tương tự, ta chứng minh được tứ giác \(AKCB\) là hình bình hành.b)    Do \(AHBC\) là hình bình hành nên \(AH//BC\), \(AH = BC\). Tương tự, \(AKCB\) là hình bình hành nên \(AK//BC,AK = BC\). Suy ra ba điểm \(H,A,K\) thẳng hàng và \(AH = AK\). Vậy \(A\) là trung điểm của \(HK\).

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